The walk that interference can stop

The cloud light has to walk through followed photons scattered many times through a slab and found that they do not cross it; they wander, and the fraction that emerges from the far side falls not exponentially but as one over the thickness. That is Ohm’s law for light, the transport the diffusion equation describes, and it is why a cloud thirty scattering lengths thick still lets through a third of the sunlight. It treated every photon as a particle taking a random walk, and it said so: phase had been discarded.

What phase does, once it is kept, is the subject of this essay, and it is not a small correction. In 1958 Philip Anderson showed that a quantum particle moving through a sufficiently disordered crystal can stop moving altogether — not slowed, not scattered into a diffusion, but trapped in a region of fixed size, by interference between the many paths it could take. Anderson was thinking about electrons. The mechanism belongs to waves of every kind, and the cleanest place to see it is light in the simplest disordered structure there is.

Kept and thrown away #

Take a stack of transparent layers, alternating between two materials with refractive indices 1 and 2.6, each layer’s thickness scattered at random by half its value either side of a quarter wavelength. Every surface reflects part of the light. There are two ways to add up what gets through.

The walk’s way adds the surfaces’ reflections and transmissions as intensities, the way a pile of glass plates was treated long before anybody thought of phase. The wave’s way carries amplitudes through every layer, keeping track of how far each partial wave has travelled, which is what multiplying the layers’ two-by-two transfer matrices does exactly.

With interference thrown away, the stacks behave like a cloud. The transmission falls as one over the thickness — checked to grow in exact proportion — and at 400 layers a per cent still gets through. With interference kept, the same stacks behave like nothing a walk can do. The typical transmission falls in a straight line on the logarithmic scale, by a factor of $e$ every 17 layers, and at 400 layers it is below one part in ten billion.

The straight line is the signature. A transmission that falls exponentially with thickness is what a barrier does, or an absorber, and this stack is neither: every layer is perfectly transparent, and no energy is lost anywhere. The light is being turned back by interference alone, the reflections from hundreds of random surfaces adding up, over any distance longer than 17 layers, to a mirror. The distance over which that happens is the localisation length.

Why one dimension always localises #

The reason is easiest to see from the walk that fails to describe it. A scattered wave can travel along any sequence of scattering events, and every closed loop in that sequence — a path that leaves a point and returns to it — can be travelled in either direction. The two directions cover exactly the same distance, pick up exactly the same phase, and arrive back at the start in step. Their amplitudes add, so the chance of returning is twice what the walk predicts for those paths, and the chance of getting anywhere else is correspondingly reduced.

That doubling is the cone of enhanced backscattering the multiple-scattering account mentioned, seen from the inside. Its effect on transport depends on how many returning loops there are, which is a question about random walks with an exact answer.

The counts come from exact lattice sums, not simulation. In one dimension a walk returns to its start over and over, and the expected number of returns grows as the square root of the number of steps, fifteen after 400. In two dimensions it grows without limit too, but only as a logarithm, 1.75 after 400. In three dimensions it levels off at 0.516: a walk in three dimensions returns a finite number of times and then escapes for good. That is the theorem George Pólya proved in 1921, which the walk that comes home drew.

Translated into waves, the counts decide everything. The interference correction to transport grows with the number of returning loops, so in one dimension it grows without limit as the sample lengthens, and any disorder at all, however weak, eventually localises the wave: the weaker the disorder, the longer the localisation length, but it is always finite. In two dimensions the correction also grows without limit, logarithmically, and a large enough sample always localises, though the size needed can be astronomically large for weak disorder. In three dimensions the correction stays finite, weak disorder only slows diffusion, and localisation requires the scattering to be strong.

The electrons Anderson was thinking about #

Anderson’s model was a crystal of atoms each holding an electron in a well, coupled to its neighbours so that an electron can hop between wells — the arrangement that makes bands and metals — with the depths of the wells scattered at random. Hopping between identical wells spreads an electron through the whole crystal. Hopping between wells of random depth mismatches every step, and past a critical amount of disorder the electron’s wavefunction stops spreading and decays exponentially away from where it started. The material stops conducting, not because its electrons are bound to atoms but because interference has confined them to regions of fixed size.

The weak version of the effect is measurable in any thin metal film at low temperature, and the way it is measured is the most direct evidence for the loop argument. The pair of paths that go round a loop in opposite directions return in step only if nothing distinguishes the two directions. A magnetic field does: an electron circling a loop one way picks up the phase that a magnetic flux leaves on a closed path, and circling the other way picks up the opposite phase. A weak field therefore destroys the extra returning, and the film’s resistance falls when a field is applied — the reverse of the ordinary increase — by an amount whose dependence on field is set by the sizes of the loops the electrons travel. That negative magnetoresistance is the fingerprint of weak localisation, and it was one of the first quantum effects found in the resistance of an ordinary wire.

A number that is not a number #

Once a medium localises, the idea of its transmission as a single number fails, and the way it fails is instructive.

The logarithms of the transmissions of 160 random stacks spread over a band roughly bell-shaped and 1.5 powers of ten wide on each side of the typical value, $6.8 \times 10^{-4}$. The plain average of the transmissions is 0.037 — 1.7 powers of ten higher, set by the few stacks at the top of the histogram. A sample picked at random almost never transmits the average. The number that describes a typical sample is the average of the logarithm, and the average of the transmission is a property of the rare samples that happen to resonate.

That is a qualitative break with diffusion. A diffusive slab has a transmission that fluctuates little from one random sample to the next, because it is the sum of very many roughly independent paths, and the grain of a speckle pattern is the only reminder of the interference beneath. A localised slab is dominated by a handful of paths, or rather by a handful of modes, and which ones it has is a matter of chance.

Resonances hidden in the disorder #

Looking at a single stack across a range of wavelengths shows what those rare transmitting samples contain.

Across most of the band the stack transmits a tiny and wildly fluctuating fraction, from a thousandth down to a millionth and back as the wavelength changes by a few per cent. At two narrow wavelengths it transmits more than ten per cent. Each of those is a resonance: a mode of the electromagnetic field trapped somewhere inside the stack, through which light can tunnel in from one side and out the other when the wavelength matches.

Carrying the field backwards from the exit through every layer shows where the light is at the strongest resonance, and it is not near either surface. It piles up to more than a thousand times the energy density of the transmitted light 62 per cent of the way through the stack, and falls off exponentially towards both faces, with the same localisation length as the average transmission. Nothing at that depth distinguishes it. The layers there are as random as anywhere else; the mode exists because that particular sequence of random thicknesses happens to form a cavity at that wavelength.

That is the difference between a localised mode and the mode that lives in a mistake. A deliberate defect in a perfect stack traps light at a designed place and a designed colour. Disorder traps light everywhere at once, at random places and random colours, in modes whose number grows with the length of the sample and whose positions no one chose — and the transmission of the whole sample at any wavelength is decided by whether one of them happens to lie within a localisation length of both faces.

Three dimensions, and thirty years of argument #

In one dimension, and in structures that confine light to a plane, localisation of light is established. Light spreading sideways through an array of optical waveguides with randomly varied properties stops spreading and stays within a fixed width, and experiments in 2007 and 2008 showed it, in two-dimensional and one-dimensional disordered lattices respectively. Sound localises in three dimensions: ultrasound in a random network of aluminium beads was reported in 2008 to show the signatures of localisation. So do matter waves: clouds of ultracold atoms released into a disordered potential made from laser speckle were reported to localise in three dimensions in 2011 and 2012.

Light in three dimensions is where the history is uncomfortable. The criterion for strong scattering, due to Abram Ioffe and Anatoly Regel, is that the scattering mean free path must shrink to about the wavelength divided by $2\pi$ — scattering so strong that a wave cannot complete an oscillation between events. Powders of materials with very high refractive index, ground to particles near the wavelength, approach it. In 1997 a powder of gallium arsenide was reported to localise near-infrared light, from a transmission that fell faster than one over the thickness. The objection, raised at once, was that absorption does the same, and gallium arsenide absorbs at that wavelength; separating the two in a single transmission measurement proved harder than expected. In 2013 time-resolved measurements through titanium dioxide powders reported a slowing of the light’s escape at long times, a signature of localisation that absorption does not mimic. Later work traced part of that signal to fluorescence from the samples, which does.

Meanwhile calculations for dense collections of point scatterers found a reason light might never localise in three dimensions, whatever the disorder. Light is a vector wave. When scatterers are packed closer than a wavelength, they couple through their near fields — fields that fall off faster than radiation and carry no energy to infinity — and those couplings open channels through which the wave can leak around the interference that would trap a scalar wave. In the models, sound and electrons localise and light does not.

A laser with no mirrors #

A medium that scatters light strongly enough to keep it inside for a long time does not need mirrors to lase. Mix a gain medium — a dye, or a powder of a laser crystal — with a strong scatterer and pump it, and the light, trapped by multiple scattering, passes through the gain many times before escaping. Above a threshold the emission narrows and brightens exactly as a laser’s does. Such random lasers were predicted in 1968 and demonstrated in the 1990s.

In the diffusive regime a random laser’s feedback is just the long walk. When the scattering is strong enough for resonances like the one in the energy-density figure to exist, those trapped modes act as tiny laser cavities, each at its own random wavelength, and the emission spectrum breaks into sharp spikes at wavelengths that change from one sample to the next — the laser counterpart of the transmission spikes in the spectrum above. The disorder that makes a sample opaque has become a set of cavities no one designed.

What the stacks leave out #

The stacks are one-dimensional. Every figure is light at normal incidence through layers, which is exactly the case in which localisation is guaranteed. A real three-dimensional sample has returning loops that can close in any direction and far fewer of them, and nothing in these figures bears on whether it localises.

Nothing absorbs. The contrast between exponential decay and absorption, which is the central experimental difficulty, cannot arise in a lossless stack. Adding absorption would make the interference-removed curve fall exponentially too, and the two would no longer be distinguishable from their transmission alone.

The disorder is in the thicknesses only. Real disorder varies the materials, the interfaces and the shapes of scatterers, and some kinds of correlated disorder — sequences with hidden order — can suppress localisation or produce transparent windows. The uniform random thicknesses drawn are the case with no hidden structure.

The averages are over tens or hundreds of samples. Statistics of a localised medium converge slowly, precisely because the transmission is dominated by rare events, and the numbers quoted would move by tens of per cent with a different set of random stacks. The straight line and the width of the spread would not.

The reflections no figure can enumerate #

The histogram and the spectrum show the transmission of stacks, and neither shows the interference that produces it. The paths whose amplitudes cancel are the uncountable sequences of reflections back and forth between random surfaces, and the transfer matrices sum all of them exactly without ever representing any one. The mechanism is visible only in its result, and in the walk counts that say how many loops there are to interfere.

Nor does the energy-density figure show time. A localised mode holds light for a long time — its narrow resonance means a long lifetime — and a pulse sent into a localised sample leaks out slowly, over a time that grows exponentially with the sample’s thickness. That slow leak is what the three-dimensional experiments looked for, and a static picture of where the light is at one wavelength is the frequency-domain half of it.

Still open: whether light can be localised in three dimensions #

Whether any three-dimensional material can localise light by disorder alone is not known. The near-field argument suggests that ordinary dense random packings of point-like scatterers cannot, and some proposals try to evade it — scatterers that respond only to the magnetic field, structures that suppress the near-field coupling with a strong static magnetic field, or disorder built into a material that already has a partial band gap, where far fewer states are available for the light to leak into. Numerical calculations support some of these, and none has produced an unambiguous measurement. What a positive result would have to show is also agreed: a slowing of transport that grows with sample size, in a material whose absorption and fluorescence have been measured and excluded.

The habit worth carrying away is the question the two curves in the first figure ask. An exponential describes what is being turned back or used up; an inverse law describes what has forgotten where it was going — and when a sample shows an exponential with nothing in it to absorb, the reflections of its random surfaces have stopped cancelling at random and started adding up to a mirror.